Monday, November 28, 2005


One of my fondest memories from my further maths lessons back in my school days occurred when one of my colleagues brought to our attention the many ways a mathematician may trap a lion. In many a surreal moment I have stared wanly into the middle distance trying to recall exactly what was in the list, and the best I could do was simply remember with fondness that it was funny. Fortunately for me I need stare wanly no longer and I may trap lions to my heart's content for Bjorn has compiled a list of many ways to do this. Thank-you Bjorn! He also shares an amusing exam answer where a student took up the ages old challenge of finding x with great success!

Also I've followed the advice of Mr Goose and started using The TeXer for making small gifs of tex online - it's very handy, but I wanted to draw your attention to the excellent host site, called The Art of Problem Solving, claiming to be the world's largest online maths community. In particular I wanted to share my pleasure at their little geometric animations in the bottom left hand corner of the site as I think they are wonderful, I particularly like this one (which I have stolen (!) the end result of from their excellent site ):

P.S. Peter Woit provides yet another excellent link to online lectures, this time by Penrose, Weinberg, Maldacena amongst others at the Perimeter Institute. Peter was referring, in particular, to the emergence of space-time talks which can be found at the bottom of the list on the left, and one of the talks is by Seth Lloyd, the views of whom have been recently commented on by Lubos Motl.

My first journal club

The enthusiastic group of postdocs here at King's have organised a journal club for all of us students and interested faculty. I've not been involved in one before and so it will be interesting to see whether or not we make progress as a group or not. Our topic to guide our reading is gravitational thermodynamics and hopefully we will look at attractors as well as more recent alpha' corrections to the entropy formula. But to start off today we began with a background review. Here was our suggested reading list (totalling 315 pages)
  • Black hole thermodynamics by Simon F. Ross
  • TASI lectures on black holes in string theory by Amanda W. Peet
  • The Thermodynamics of Black Holes by Robert M. Wald
  • Black Holes by Paul K. Townsend
    An abundance of middle initials, which bodes well. We really went over some definitions, such as surface gravity for the Schwarzschild black-hole, null hypersurfaces, temperature, Hawking radiation (which we hope to look at more closely next week), event horizon area increase (and we drew the analogy with entropy, but no more than that) and the formula at the heart of black hole thermodynamics:
    So we didn't get too far into our review in an hour, and already there was some disagreement in our club between those who want to push on towards the string theory point of view and the recent papers and those who want to get the fundamental concepts under control before moving on. I foresee a rocky ride ahead, but it will be a fun endeavour. So, for now, our journal club rides on...
  • Saturday, November 26, 2005

    Healthy Scepticism in the Ranks

    Yesterday Andreas Recknagel gave our first mathematics colloquium of the year. We heard a very rapid overview of unification, string theory and conformal field theory that finished up with T-duality and mirror symmetry. He even showed us some "strings" he had in his pockets to describe the winding number. The talk was peppered with good humour the first occurred at the outset where he described "the string factory" to us, here are my notes on it:

    Heterotic Geometries, all of them

    This week George Papadopoulos talked to us about the results from his latest paper, The spinorial geometry of supersymmetric heterotic string backgrounds, with Ulf Gran and Philipp Lohrmann, from my office. I used to keep a pink penguin on the site as a tribute to Philipp, that one could poke and push off some ice while reading the blog. Happy days. Now I have to treat him with much more respect since he and his collaborators have made use of the fundamental procedure of the spinorial geometry programme (which I will describe below) to produce a list of all the possible background geometries for the heterotic string, up to multiplication by a compact group in some cases.

    The talk itself began a little ominously on Wednesday when it was realised that our electronic projector was not available, and instead of us all crowding around George's laptop, he managed to give us a blackboard talk which was impressive both for its clarity and in that he hardly ever looked to his laptop for guidance. However one consequence of this was that hardly any complex equations appeared even though there must have been many more intended, and also the speed of the talk was very good for those taking notes. I wonder if this policy of losing the projector is not something we should take up full time at King's...

    George began by motivating the study of the Killing spinor equations by looking at the Euclidean instanton bound. In 4-dimensions,
    Where we have completed the square. The bound is saturated when,
    When we take the minus sign in the above we have the self-duality conditions on the field strength. Now suppose we wanted a spinorial version of this construction. We would start by coupling the spinor to the field strength forming:
    Reproducing the equivalent of the Euclidean action above we have another formulation for the instanton bound, but now in terms of spinors,
    Normalising the bound is attained if, These are the supergravity Killing spinor equations arrived at by considering the instanton bound.

    Having motivated the Killing spinor equations, George moved on to describing the formalism he has developed for helping to solve them. He took us through a 4-dimensional example so we could see the general approach. In 4-dimensions the spin representation is Spin(4) which is the double cover of SO(4), the Euclidean rotation group. There is an isomorphism Spin(4)=SU(2)xSU(2) and there are two different Weyl spinors, with two components in each, which transform under only one of the SU(2)'s. George then described the setting up of a real vector space, whose components will act as a space of forms. In two-dimensions we have two basis elements in the vector space (e_1,e_2). Mimicking the splitting of the Weyl spinors to transform under two different copies of SU(2), we set up a second copy of this vector space by "complexifying" it. The basis of forms is:
    We have a basis of two even and two odd forms which corresponds to two different chiralities. If we explicitly write down the gamma matrices using our basis we can get some work done,
    I have used a vee for the inner derivative which acts in the opposite way to the exterior derivative; it's destructive while the wedge is constructive, e.g. . These are all we need to solve the Killing spinor equations. We first note that the equation is unchanged under spin(4) transformations, upto a Lorentz transformation on the field strength. That is we may orient the spinor how we wish, in particular we may pick the direction 1 (the "Clifford vacuum" if you like) in the basis:
    Now we may go back and look hard at our basis for the gamma matrices and write down the combinations that annihilate the vacuum. We use these to define a new (solution) basis:
    We also have a set of antiholomorphic gamma matrices in this basis which are the complex conjugates of the above. Finally expanding out the Killing spinor equation, by summing over holomorphic and antiholomorphic indices, in this basis leads to some simple expression,
    Recalling that the n index gamma matrices are just the antisymmetric combination of the n individual gamma matrices, we see immediately that the first term annihilates the Clifford vacuum. Carrying on thinking about this in terms of a basis of states is fruitful since we also see that the remaining two terms must vanish independently, since they create different "Clifford states". That is the Killing Spinor equations reduce to,
    So we see from this walkthrough just how simple the equations can become. In particular one can learn about the geometry of a number of setups. In the latest work, linked above, the authors make use of the fact that the heterotic string background resembles a Riemannian manifold, to find all the stability subgroups for the possible numbers and types of Killing spinors. From which the geometry of the background is determined. So impressively it is claimed that all the M-theory geometries corresponding to the heterotic string regime without alpha' corrections have been found. Of course it is not clear what happens to the Killing spinor equation when alpha' corrections are included, but it seems like they will be altered.

    Friday, November 18, 2005

    SuSy backgrounds and M-theory corrections

    It would be remiss of me not to point out the Vega Science Trust which has been recently highlighted on cosmicvariance by Mark Trodden; if nothing else go and watch the Feynman lectures available there. Elsewhere Seed magazine seems to be getting much blog support (see here, here and here). So jumping on the bandwagon I'd like to recommend the Seed magazine podcast so you can have "science is culture" articles wherever you take your mp3 player. In fact if you are keen on podcasts you should also listen to Berkley Groks.

    This week Kellogg Stelle from Imperial College visited KCL to tell us about his work on corrections to M-theory at order {\alpha'}^3. In particular he described work which demonstrated how supersymmetry may be preserved by making use of a corrected killing spinor equation. Indeed one may work backwards and start with a corrected killing spinor equation and rediscover the corrections to the string background. The methods used for various supersymmetric backgrounds are based on the following papers written variously with Lu, Pope and Townsend:

  • Higher-Order Corrections to Non-Compact Calabi-Yau Manifolds in String Theory (Kahler manifolds)
  • Supersymmetric Deformations of G_2 Manifolds from Higher-Order Corrections to String and M-Theory (G_2 manifolds)
  • String and M-theory Deformations of Manifolds with Special Holonomy (Spin_7)
  • Generalised Holonomy for Higher-Order Corrections to Supersymmetric Backgrounds in String and M-Theory (Generalised holonomy)

    If you want to learn more read through the first paper above, there are some surprising results. You might even be keen on thinking about the alpha' corrections and the group E10, or even their effect on entropy calculations. I'd write more but 'tis the season to be making postdoc applications and I really have to be seasonal, very quickly and as many times as possible. Ug, the drudgery.
  • Wednesday, November 09, 2005

    A Silver Age in Black Hole Research?

    Yesterday afternoon Harvey Reall from Nottingham University came to talk at Imperial College. The talk entitled Black Holes and Extra Dimensions was aimed at masters level students, of which, there was only one in the audience :( However it was a nice talk, and somewhat of a black hole history lesson and worth describing here. Lubos Motl has a report of a very similar talk by Reall which you can find here.

    Harvey began by describing the "Golden Age" (so named by Kip Thorne) of black hole research which began in 1963 with the discovery of the Kerr solution, and supposedly lasted until 1973 when the macroscopic black hole entropy formula was discovered by Hawking. The two main achievements of the "age" according to Reall were:
    1. The black hole uniqueness theorems
    2. Black hole thermodynamics
    The uniqueness theorems (for more information you can read David Robinson's "Four decades of black hole uniqueness theorems")in 4 dimensions are twofold; there are separate proofs for the non-rotating and the rotating vacuum solutions. The non-rotating solution is due to Werner Israel (1967) and a more recent proof by Bunting and Masood-ul-Alam (1987) and shows that the only non-rotating, equilibrium solution is the Schwarzschild solution (1917). The only rotating, equilibrium solution is the Kerr solution (1963) as shown by Carter (1971), Hawking (1972) and Robinson (1975). These solutions are described by one (M) and two (M,J) parameters respectively. Generalisations to include the Maxwell field also exist (the Kerr-Newman solution).

    Reall spent some time motivating the consideration of extra dimensions. For those interested in string theory he simply said that it is, in his opinion, the best candidate for a theory of quantum gravity. He justified this by saying that the entropy macroscopic entropy formula is the only experimental data that we have for quantum gravity and that microscopic count of entropic degrees of freedom gives matching numbers. For those not so convinced by string theory Reall offered the AdS/CFT (Maldacena 1997) or as he called it the Gauge/Gravity correspondence. The gauge/gravity correspondence was described for the case of a 5D gravity theory in the interior of the cylinder (shown right) being equivalent to a 4D gauge theory on the curved surface of the cylinder - the edge of the cylinder is at infinity by some clever choice of coordinates and three dimensions have been suppressed out of respect for our 4 dimensional universe. In short Reall pointed out that some 4D gauge theories are equivalent to 5D gravitational theories and so we may learn more about QCD calculations by looking at higher dimensional theories.

    Reall then moved on to talk about constructing 5D solutions from the known 4D solutions. He began with the black string which was constructed by adding an extra flat dimension to the Ricci flat 4D Schwarzschild solution: This is the black string, it is infinitely long and so has infinite energy. To avoid this we compactify the z direction by identifying z~z+2/piR, the geometry is changed to 4D Minkowski space crossed with a circle. Reall told us that the Black string This is the black string, it is infinitely long and so has infinite energy. To avoid this we compactify the z direction by identifying z~z+2/piR, the geometry is changed to 4D Minkowski space crossed with a circle. Reall told us that the Black string is classically unstable when the radius of compactification, R, is greater than 2M. This is the so-called Gregory-Laflamme instability (1993) and it's endpoint is unknown. However Horowitz and Maeda (2001) conjectured that the endpoint was the non-uniform black string, which has a sine-wave distortion of the event horizon as it moves through its z coordinate. Toby Wiseman showed that these exist using a numerical approach in 2002. The downside is that they lead to decreasing entropy, which is unrealistic!

    There is a generalisation of the Kerr solution to D dimensions called the Myers-Perry solution (1986). It has some familiar properties in that it's event horizon has topology S^{D-2} and it is uniquely specified by its mass and its angular momenta (there being [(D-1)/2] angular momenta, where [] means drop the fractions). Another familiar property is that the angular momenta are bounded. Having described this much Harvey Reall asked if there were any other types of higher dimensional black holes, i.e. does the rotating black hole uniqueness theorem still hold in higher dimensions?

    Harvey answered that he and Emparan had showed that there existed a different type of black hole in 5 dimensions (2001) - the black ring. This solution is a rotating closed loop of black string, whose gravitational collapse is held at bay by one of its angular momenta. Heuristically the gravitational force is balanced by the centrifugal force. Of course one wonders why a similar rotating black cylinder couldn't exist in project to 4D one could set r=0, corresponding to zero mass, hence no 4D equivalent solution to the black ring, at least not in this way. The 5D black ring's second angular momentum is zero and the solution has only 2-parameters with topology S^1 cross S^2.

    It transpires that for a certain range of angular momentum there are two ring solutions, one small and one large. So together with the Myers-Perry solution there are some regions of parameters where 3 solutions exist for the given parameters. The uniqueness theorem for rotating black holes is well and truly lost for higher dimensions. The non-rotating uniqueness solution does still hold in higher dimensions (Gibbons, Ida & Shiromizu 2002).

    Harvey finished up by telling us about supersymmetric black rings (also known in some circles as: "The Hairy, Tiny Black Hole Donut Theory") which were discovered by Elvang, Emparan, Mateos and Reall (2004); Gauntlett and Gutowski (2004); and Bena and Warner (2004). You can listen to Reall talk about black rings here. The solution is now stabilised by a non-zero second angular momentum (this one rotates the torus about its second circle). The solution has 7 parameters: 2 angular momenta, 3 electric charges, 3 magnetic dipoles and one relation between them all. Also noteworthy is the observation that the dipoles are not conserved, so there are only 5 conserved quantities amongst the seven parameters. The entropy formula is consequently much more complicated than usual but despite this it seems that string theory can still be used to give a correct microscopic count of the solutions degrees of freedom (Cyrier, Guica, Mateos and Strominger 2004), although Reall had some doubts about the count since it seemed to imply that one of the two angular momenta was zero, contrary to the macroscopic solution.

    In his final remarks Reall reminded us that the golden age of black hole research was started off by the discovery of the Kerr solution, and he hoped that the silver age would be kicked off by the discovery of the black ring. Personally I'm already worried about naming the third age, since the bronze age is already taken.

    Friday, November 04, 2005

    A Bridge to Higher Spin Theory and AdS/CFT

    Last weekend I was pleasantly surprised when a friend phoned me and said he was in my neighbourhood and thought we could go for a drink. Unfortunately there were scheduled works on a bridge between where he was and where I live; I thought the bridge would be raised and he replied asking if that meant we were topologically disconnected... This was somewhat similar to how I felt yesterday at our weekly seminar, since the topic seemed to be "topologically disconnected" from my own small area of understanding. So in these comments I propose to try to build a small bridge to the beginnings of the subject matter discussed in the talk.

    So Paul Heslop from DAMTP in the real Cambridge came to talk to the King's group under the title "On the higher spin/gauge theory correspondence" which is based on his work with M. Bianchi and F. Riccioni. Paul's talk was based mostly around their paper "More on La Grande Bouffe: towards higher spin symmetry breaking in AdS". Don't be put off by "La Grande Bouffe" it is the name of a film and it refers to an aspect of the theory where a higher spin field "eats" the entire chain of lower spin fields to acquire mass, these chains can be very long. The film of the same name, we were told by an anonymous member of our department, is about "three men and one woman" and the men all eat so much that they die! Via IMDB the plot is described as:
    Four successful middle-aged men Marcello, a pilot; Michel, a television executive; Ugo, a chef; and, Philippe, a judge go to Philippe's villa to eat themselves to death.
    It has 7.2 stars from 1225 reviews. So don't be put of by the term, which I believe is due to Massimo Bianchi.

    The conjectured correspondence is that "the massless higher spin theory is holographically dual to a free gauge theory on the boundary". We can summarise it best using a picture given in Paul Heslop's talk:
    If you wish to learn about the correspondence then you can read through the following papers of Bianchi et al:

  • "Higher Spin Symmetry and N=4 SYM" by Niklas Beisert, Massimo Bianchi, Jose F. Morales and Henning Samtleben
  • "Higher spin symmetry (breaking) in N=4 SYM theory and holography" by Massimo Bianchi
  • "Higher Spins and Stringy AdS5xS5" by Massimo Bianchi

    However since there is a gap in my understanding, I thought it would be constructive to find a brief paragraph or two to express the formative ideas behind Vasiliev's higher spin theory, if I can. However a better option would be to listen and see pictures from a talk by Vasiliev here, where you will hear him commence by describing the totally symmetric massless free fields of Fronsdal (1978) and de Wit and Freedman (1980) where the bosonic case contains a field with s symmetrised indices which is double-traceless (i.e. if you contract two pairs of its indices it is zero). There is a uniquely associated action which is chosen by requiring that some action containing terms up to second order in derivatives of the field is gauge invariant. There are some familiar gauge invariance principles for s=1 (Maxwell) and s=2 (gravity) fields and the essence of higher spin theory is the question of whether there is a unifying gauge symmetry that exists for other higher spin fields as well.

    What are the connections with strings and supergravity I hear you cry? Well Vasiliev goes on to say (in the video above) that while there is a limit on the spin of the massless particles in a d-dimensional theory (s=d-2) there is also a corresponding limit on the number of supersymmetries (e.g. he draws a parallel between the s<=2 and N<=8 in 4-dimensions). He says that this is "practically equivalent" to the limit on the number of dimensions of supergravity (d=11) and says that if one wishes to consider theories beyond supergravity then one might start by wondering what happens when one includes massless spin fields of spins higher than those in D=11 sugra. Further motivation comes from the Stueckelberg symmetries of the superstring which are similar to spontaneously broken symmetries of higher spin gauge symmetries. As well as from the work of Sundborg and Witten, arguing for a nonlinear theory with infinite higher spin fields in the bulk.

    So there you have it a small bridge to travel over and go and study higher spin gauge theory. A couple of further papers to help you on your way towards the triangle above are:

  • "Higher Spin Gauge Theories"(be warned this is a link to a 205 page pdf), Volume 1 of the proceedings of the Solvay Workshop, a compilation of work in various areas related to higher spin theories.
  • "Notes On Higher Spin Symmetries" by Andrei Mikhailov

    It transpired last weekend that the scheduled roadworks were not happening, the bridge was down, and just like this week's seminar experience my friend and I were topologically connected after all, and we went and had a merry afternoon in Greenwich.